This is my first post, so sorry if this is a really novice and idiotic question.

I followed the setup assistant with my new 27" imac, just to make sure the computer was working. Then, later I followed Migration Assistant to move everything from my 2009 MBP. This landed me with two users. I deleted the one set up with setup assistant on the advice of a computer nerd friend. But now, I realize, while I have an icon for the old hard drive on my desktop, I don't know where the hard drive for the new imac is. I thought it was the same as the old "Macintosh HD", but that one won't let me put anything on it, so I'm figuring it's really the old one. Can anyone set me straight?

Normally Macintosh HD, especially the root directory is not used to store files on, that is what your home directory is there for.
You can also navigate to your Macintosh HD via pressing CMD+SHIFT+C in Finder and clicking on Macintosh HD or select hard disks to show up on the Desktop via Finder > Preferences > General > Show these items on the Desktop or select hard disks to show up in the Sidebar under Devices via Finder > Preferences > Sidebar > Devices > Hard disks.

Should I be seeing something called "iMacintosh System" somewhere? I don't see that anywhere. However, I did check permissions on the desktop and "Macintosh HD" folder and I should have permission to read and write. I just can't see where to put stuff on this computer's HD. About to see about the other things you mentioned...

Macintosh HD is the old drive from the old computer, right? Shouldn't M's iMac be the drive for the new computer? I can't even see where to save files to the new imac's hard drive. And I can't save files to the Macintosh HD either, even though I should have permissions. When I try to drag and drop a folder to Macintosh HD, it's telling me "The item “a” can’t be moved because “Macintosh HD” can’t be modified."

Okay, Macintosh HD is the standard name of every Mac's HDD, if I recall correctly (I never purchased a Mac with an SSD already built into it).
"Molly's iMac" is the name of the computer, not the HDD.
The name of the computer can be changed via System Preferences > Sharing > Computer Name to anything you want.

The same goes for Macintosh HD, via Finder > CMD+SHIFT+C (the shortcut to "Molly's iMac" - now you arrive at an overview of all connected internal and external storage devices, like the Macintosh HD, a USB flash memory thumb drive or any other external HDD or internal/external optical disk drive (ODD)(if a medium is inside that ODD)) > Macintosh HD > right click on it > Get Info > Name & Extension.
Thus everything is working on your Mac as it should, you do not save files to "Molly's iMac" (it is an overview of connected devices), but to "Macintosh HD" (though it is not recommended to do so, as you have other places to store files, and recently I helped a forum member, who did save files/folders in her Applications folder and on Macintosh HD, which was just a mess).

Your home directory (the user folder for the active account, in your case the account is "MVP") is located at Macintosh HD / Users / MVP and can be accessed via Finder > Sidebar > Favorites > MVP or Finder > CMD+SHIFT+H.
In there you see a couple of preset folders (Documents, Desktop, Music, Pictures, Movies, ...), which one normally uses to store files in, or one creates one's own structure within that home directory.

Thank you so much. That helps tremendously! Sorry I was so confused about something that seems ridiculously simple. I don't necessarily want to save a file to the HD now, but I do sometimes store temporary files on the HD, rather than the desktop, since way back in the day I was told by a computer guy at a job not to store stuff on the desktop. My most valuable files, I save on externals. I didn't know I should be saving stuff to those preset folders.